reserve C for CategoryStr;
reserve f,f1,f2,f3 for morphism of C;
reserve g1,g2 for morphism of C opp;
reserve C,D,E for with_identities CategoryStr;
reserve F for Functor of C,D;
reserve G for Functor of D,E;
reserve f for morphism of C;

theorem Th41:
  for A being Category, a1,a2 being (Morphism of A),
  f1,f2 being morphism of alter(A) st a1 = f1 & a2 = f2 &
  [a1,a2] in dom the Comp of A holds a1(*)a2 = f1(*)f2
  proof
    let A be Category;
    let a1,a2 be Morphism of A;
    let f1,f2 be morphism of alter(A);
    assume
A1: a1 = f1 & a2 = f2;
    assume
A2: [a1,a2] in dom the Comp of A;
    thus a1(*)a2 = (the composition of alter A).(f1,f2) by A1,A2,CAT_1:def 1
    .= f1(*)f2 by A2,A1,Def2,Def3;
  end;
